Quantum AG code[1]
Description
A Galois-qudit CSS code constructed using two linear AG codes.Rate
Quantum AG codes [1] can be asymptotically good. There exist three such families [2–4] that admit a diagonal transversal gate at the third level of the Clifford hierarchy.Magic
By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum AG codes that admits a diagonal transversal gate at the third level of the Clifford hierarchy and attains a zero magic-state yield parameter, \(\gamma = 0\) [2]. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see [5,7][6; Sec. 5.3]. Two other such asymptotically good families exist [3,4], admitting a different diagonal gate at the third level of the Clifford hierarchy.Encoding
Encoding defined in Ref. [1] uses a technique from Ref. [8] to encode quantum stabilizer codes.Transversal Gates
There exist three asymptotically good code families [2–4] that admit a diagonal transversal gate at the third level of the Clifford hierarchy.Cousins
- Algebraic-geometry (AG) code
- Triorthogonal code— By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum AG codes that admits a diagonal transversal gate at the third level of the Clifford hierarchy and attains a zero magic-state yield parameter, \(\gamma = 0\) [2]. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see [5,7][6; Sec. 5.3]. Two other such asymptotically good families exist [3,4], admitting a different diagonal gate at the third level of the Clifford hierarchy.
- Tsfasman-Vladut-Zink (TVZ) code— The AG codes used in an asymptotically good construction of quantum AG codes with non-Clifford transversal gates [3] are those of the TVZ codes.
- Galois-qudit GRS code— Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction from GRS codes, which are evaluation AG codes.
Primary Hierarchy
Parents
Quantum AG codes can be realized in the CSS code construction [9].
Quantum AG code
Children
Galois-qudit RS codes are constructed via the CSS construction from RS codes, which are evaluation AG codes.
References
- [1]
- R. Matsumoto, “Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes”, IEEE Transactions on Information Theory 48, 2122 (2002) arXiv:quant-ph/0107129 DOI
- [2]
- A. Wills, M.-H. Hsieh, and H. Yamasaki, “Constant-Overhead Magic State Distillation”, (2024) arXiv:2408.07764
- [3]
- L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
- [4]
- Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140
- [5]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [6]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
- [7]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [8]
- A. Ashikhmin and E. Knill, “Nonbinary Quantum Stabilizer Codes”, (2000) arXiv:quant-ph/0005008
- [9]
- A. Niehage, “Nonbinary Quantum Goppa Codes Exceeding the Quantum Gilbert-Varshamov Bound”, Quantum Information Processing 6, 143 (2006) DOI
Page edit log
- Victor V. Albert (2024-08-23) — most recent
Cite as:
“Quantum AG code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_ag